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Radial Velocity and Proper Motion
Barbieri Chapter 9
Kathy Geise
Stellar or Annual Parallax
• The apparent motion of a
nearby star is a small ellipse in
the sky relative to background
stars over the period of a year.
• The parsec is the distance for
which the annual parallax is
one arcsecond.
• The reciprocal of the parallax
is the distance in parsec.
Two Components of Velocity
Over time, star X moves to position X’ as seen from the heliocenter S along
velocity vector, V. The velocity vector has components vr in the radial
direction and vt in the transverse direction.
Proper Motion, μ
The linear drift in this graph is from
proper motion. The elliptical motion is
from the annual parallax.
A slow, angular change to
equitorial coordinates due to a
star’s transverse velocity
relative to the viewer (Carroll
and Ostlie) or heliocenter
Units of arcseconds per year.
Tangential velocity vt is related
to proper motion, μ, as follows:
μ = π/4.740 *vt (arcsec/yr)
where π is annual parallax in
arcsec and vt in km/s.
Barnard’s star
The Sky 6
• Second closest to Sol after
Alpha Centauri
• 6.0 light years away
• Constellation Ophiuchus
• Dim red dwarf (m. 9.55)
• Old star, possibly 11-12 by
• Approaching Sol
• Largest known proper motion
of all stars (10.3”)
Radial Velocity
• Radial velocity is the
component of the velocity
toward or away from the
• Radial velocity is measured
through the Doppler effect.
• When the velocities measure
less than 0.01c, we can use
prerelativistic formulae:
– Vr = c*(λobserver – λs)/ λsource
– z=Δλ/λ=Vr/c
• Z>0 red-shifted; radial
velocity positive
• Z<0 blue-shifted; radial
velocity negative
Astrometric Velocities
• “Precise astrometry permits accurate determination of stellar radial
velocity from purely geometric measurements without using
spectroscopy nor employing the Doppler principal.”
• Changes in parallax and proper motion over time reveal radial
• Possible only with precise measurements.
• Hipparcos satellite data in combination with older stellar data.
• Changing angular separation of stars sharing the same space
velocity as in a moving cluster measured by Hipparcos.
• View a Java animation of Bernard’s Star proper motion on the
Hipparcos website:
– The Hipparcos and Tycho Catalogues: Nearby Stars
Hipparcos animations
Animation of intermediate astrometry data of
Hipparcos star.
Motion of the Pleiades over
120,000 years.
This is an 8x6 degree field.
Relativistic Considerations
• The whole velocity vector is affected by relativity, not just
the radial component. Relativity predicts a transverse
Doppler effect.
• General relativity is important for high accuracy, for
example, spacecraft moving within the solar system.
• High velocity stars, like Barnard’s
• Expanding gaseous envelopes of exploding supernovas
• High red-shifted, very distant galaxies
• Light emitted from atoms in a gravitational field, for
example the surface of a star, may be red-shifted.
The Sky 6
Snake Nebula
Cover photo APOD Snake Nebula